Comment: This paper
was written in 1988. It only outlines certain circles of ideas. Later on I further
developed and explained these ideas in seminars given in different places,mostly
because of interest and encouraging comments from Daniel Kastler and Robert
Coqueraux in Marseille. One of the references refers to an "unpublished
paper" by Detlev Buchholz and myself. I still have the manuscript with
the added name of Robert Olkiewicz, who contributed important computations.
It is still unpublished  waiting for a final touch. Perhaps, one day, I will
fill in the gaps and put it on the web.... It is not a bad paper.
I would also like to comment on the interplay between classical and quantum
discussed briefly below. As early as 1988 I was developing this idea in collaboration
with Philippe Blanchard (Bielefeld). It grew into a new theory: Event Enhanced
Quantum Theory (EEQT), with dozens of published papers and conference talks
which resulted. In short: this is a quantum theory of an individual quantum
system. No "ensembles" are ncessary. It tells us how Nature does what
it does. Or at least it sets a hypothesis as to how Nature is doing what it
is doing. For more on this subject see EEQT papers in my list of publications.
On Berry's phase^{1}
A. Jadczyk
Institute of Theoretical Physics, University of Wroclaw
Abstract: We discuss geometroalgebraic aspects of the Berry's phase phenomenon. In particular we show how to induce parallel transport along states via KaluzaKlein mechanism in infinite dimensions.
1. Classical and Quantum Worlds
In recent times, an increasing amount of work is being devoted to developing a new mathematical framework of noncommutative geometry, also called quantum geometry. As a result, we have:
So, it becomes natural to ask this question: is everything going to be quantum? A monist would probably answer this question "yes, that is our lot, the 'true' description is the quantum description ...".
Being neither a monist nor a dualist, I would like to reiterate the well known and so often repeated position taken by Niels Bohr: that there is a cut between the classical and the quantum; a necessary cut; where classical is, roughly, everything that can be expressed in terms of an ordinary language; which can be communicated.
Thus we have the following:
Dualistic Picture


Quantum

Classical

Matter

Language
(= Knowledge)

Today, this is nothing but a picture. What we need is a theory, a theory that will make the picture mathematically precise and quantitatively predictive. The dualism we have in mind is similar to the one we know from General Relativity, namely the dualism between Matter and Geometry. There is no better way to express its idea than as is done by John Archibald Wheeler, who puts the story into these words:
An analogous deep and simple statement concerning the picture above is still lacking^{ 2} What we need, in particular, is an adequate mathematical theory of the substance of (objective) knowledge  then a statement such as "the wave function describes state of knowledge" would have a precise meaning, devoid of any subjectivity. Although it is not directly related to our subject, perhaps it is, worth our while to note that the picture above necessarily implies that we should deal, as strongly stressed by Prigogine,with open systems.^{3}
2. Classical Geometry of the Quantum Space
In quantum theory one associates operators with yesno questions and with observables. However the question "what is the state of the quantum system" is NOT a quantum question. It pertains rather to the right hand side of the picture above, to the classical domain. Of course, one could think of some "quantum metasystem", where the question of the state of a "system" would be represented by an operator (or 'superoperator')  but then we could ask of the state of the metasystem, metameta system, etc., ad infinitum. Until a selfconsistent theory that closes this infinite chain ^{4} is established, we have, in the quantum framework, a distinguished classical object: the State Space of the quantum system. Much of the properties ^{5} of the system are determined by geometry of this space. Here we would like to discuss only one topic related to this geometry: parallel transport along states. It is convenient, for doing this, to use the algebraic language.
...............................................
2.1 Observables and States
2.2 GNSconstruction
2.3 The fibration P
3. Example: the case of A=B(H)
4. Berry's phase
Download Paper in PDF format (160 kB)
Footnotes
1. Based on talk presented at the 7th Summer Workshop on Mathematical Physics  Group Theoretical Methods and Physical Applications. Clausthal, Germany, July 1988
2. Cf. however the idea of interaction between World I and World III by Popper and Eccles
3. I am indebted to Rudolph Haag for an illuminating discussion on this subject
4. Some ideas in this direction can be found in Wheeler , see also Rössler
5. According to Mielnik, perhaps even all properties of the quantum system
References
Popper, K.R., and Eccles, J.C., The self and its brain. An argument for interactionism, Routledge and Kegan Paul, London and New York, 1986.
Prigogine, I., From being to becoming. Time and complexity in the physical sciences, W.H. Freeman and Co, San Francisco, 1980.
Wheeler, J.A., Bits, Quanta, meaning, in Problems in Theoretical Physics, ed. A. Giovannini, F. Mancini and M. Marinaro, University of Salerno Press, 1984.
Rössler, O.E., Endophysics, in Real Brains  Artitficial Minds, ed. J.I. Casti and A.. Karlqvist, North Holland, New YorkAmsterdamLondon, 1987.
Mielnik, B., Commun,Math.Phys. 37 (1974) 221256.
Dixmier, J., C*Algebres et leurs representations, GauthierVillars, Paris, 1969.
Uhlmann, A., Parallel Tramsport and Holonomy along Density Operators, in Proc. of the XV Int. Conf. on Differential Geometric Methods in Theoretical Physics, ed, H.D. Doebner and J.D. Hennig, World Scientific, SingaporeNew JerseyHong Kong, 1987.
Uhlmann, A., Parallel Transport and "Quantum Holonomy" along Density Operators, Rep. Math.Phys. 24 (1986) 229240.
Van Daele Alfons, Celebration of Tomita's Theorem, in Proc. of Symp. Pure. Math. 38(1982), Part 2, 14. \bibitem{buchjad}
Buchholz, D. and A. Jadczyk, to appear
Berry, M.V., F.R.S., Quantal Phase phactors accompanying adiabatic changes, Proc. Roy. Soc. London A 392 (1984)
Simon, B., Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase, Phys.Rev.Lett., 51 (1983 ), 21672170.
Wilczek, F., and A.Zee, Appearance of gauge structure in Simple Dynamical Systems, Phys. Rev. Lett. 52 (1984), 2111 2114
Aharonov, Y., and J. Anandan, Phase change during a cyclic quantum evolution, Phys. Rev. Lett. 58 (1987) ,15931596.
.